OFFSET
0,6
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
EXAMPLE
The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
(53) (72) (73) (B) (75) (D) (B3)
(621) (532) (A1) (651) (B2) (752)
(4221) (631) (4331) (732) (A21) (761)
(4411) (6321) (43321) (A31)
(6411) (44311) (C11)
(43221) (6521)
(44211) (9221)
(54221)
(64211)
MATHEMATICA
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n], Times@@red/@#==red[Times@@Prime/@#]&]], {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
STATUS
approved